   Chapter 11.1, Problem 49E

Chapter
Section
Textbook Problem

Determine whether the sequence converges or diverges. If it converges, find the limit. a n = ln ( 2 n 2 + 1 ) − ln ( n 2 + 1 )

To determine

a) The sequence converges or diverges

b) If it converges, find the limit

Explanation

1) Concept:

Use the definition, and by using theorem and limit laws, find the limit.

2) Definition:

Converges: If limnan exists then the sequence is called convergent

Diverges: If limnan does not exists then the sequence is called divergent

3) Limit laws:

a) limnanbn=limnanlimnbn

b) limnlnn=

4) Theorem:

If limnan=L anf the function f is continuous at L, then limnf(an)=f(L)

5) Given:

an=ln(2n2+1)-ln(n2+1)

6) Calculation:

Consider the given sequence,

an=ln(2n2+1)-ln(n2+1)

Find limit as n

Therefore,

limnan=limnln(2n2+1)-ln(n2+1)

Simplify,

=limn

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